Source distribution in interferometry for wave and diffusion

For the wave equation, the Green’s function that describes the wave propagating between two receivers can be reconstructed by cross-correlation if the receivers are enclosed by sources on a closed surface. This technique is normally called interferometry. The ordinary operator used in this technique is cross-correlation. The same technique for Green’s function extraction can be applied to the solution of the diffusion equation if there are sources throughout in the volume. In practice, we only have a finite number of active sources. We address the question what minimum source density is needed for the accurate extraction of the Green’s function, and how these sources should be located on the source in the wave problem and if it is possible to reconstruct the Green’s function of the diffusion equation by using a limited number of sources within a finite volume. We study these questions for homogeneous and isotropic media for both wave propagation and diffusion using numerical simulations. These simulations show that for the used model, the angular distribution of sources is critical in wave problems. For diffusion, the sensitivity of the sources decays away from the center of the two receivers. The required width of the source distribution decreases with frequency, and therefore the required source distribution for early time and late time reconstruction is different.